On the Hardness of Partially Dynamic Graph Problems and Connections to Diameter

Conditional lower bounds for dynamic graph problems has received a great deal of attention in recent years. While many results are now known for the fully-dynamic case and such bounds often imply worst-case bounds for the partially dynamic setting, it seems much more difficult to prove amortized bounds for incremental and decremental algorithms. In this paper we consider partially dynamic versions of three classic problems in graph theory. Based on popular conjectures we show that: -- No algorithm with amortized update time $O(n^{1-\varepsilon})$ exists for incremental or decremental maximum cardinality bipartite matching. This significantly improves on the $O(m^{1/2-\varepsilon})$ bound for sparse graphs of Henzinger et al. [STOC'15] and $O(n^{1/3-\varepsilon})$ bound of Kopelowitz, Pettie and Porat. Our linear bound also appears more natural. In addition, the result we present separates the node-addition model from the edge insertion model, as an algorithm with total update time $O(m\sqrt{n})$ exists for the former by Bosek et al. [FOCS'14]. -- No algorithm with amortized update time $O(m^{1-\varepsilon})$ exists for incremental or decremental maximum flow in directed and weighted sparse graphs. No such lower bound was known for partially dynamic maximum flow previously. Furthermore no algorithm with amortized update time $O(n^{1-\varepsilon})$ exists for directed and unweighted graphs or undirected and weighted graphs. -- No algorithm with amortized update time $O(n^{1/2 - \varepsilon})$ exists for incremental or decremental $(4/3-\varepsilon')$-approximating the diameter of an unweighted graph. We also show a slightly stronger bound if node additions are allowed. [...]Comment: To appear at ICALP'16. Abstract truncated to fit arXiv limit

Finding Even Cycles Faster via Capped k-Walks

In this paper, we consider the problem of finding a cycle of length $2k$ (a $C_{2k}$) in an undirected graph $G$ with $n$ nodes and $m$ edges for constant $k\ge2$. A classic result by Bondy and Simonovits [J.Comb.Th.'74] implies that if $m \ge100k n^{1+1/k}$, then $G$ contains a $C_{2k}$, further implying that one needs to consider only graphs with $m = O(n^{1+1/k})$. Previously the best known algorithms were an $O(n^2)$ algorithm due to Yuster and Zwick [J.Disc.Math'97] as well as a $O(m^{2-(1+\lceil k/2\rceil^{-1})/(k+1)})$ algorithm by Alon et al. [Algorithmica'97]. We present an algorithm that uses $O(m^{2k/(k+1)})$ time and finds a $C_{2k}$ if one exists. This bound is $O(n^2)$ exactly when $m=\Theta(n^{1+1/k})$. For $4$-cycles our new bound coincides with Alon et al., while for every $k>2$ our bound yields a polynomial improvement in $m$. Yuster and Zwick noted that it is "plausible to conjecture that $O(n^2)$ is the best possible bound in terms of $n$". We show "conditional optimality": if this hypothesis holds then our $O(m^{2k/(k+1)})$ algorithm is tight as well. Furthermore, a folklore reduction implies that no combinatorial algorithm can determine if a graph contains a $6$-cycle in time $O(m^{3/2-\epsilon})$ for any $\epsilon>0$ under the widely believed combinatorial BMM conjecture. Coupled with our main result, this gives tight bounds for finding $6$-cycles combinatorially and also separates the complexity of finding $4$- and $6$-cycles giving evidence that the exponent of $m$ in the running time should indeed increase with $k$. The key ingredient in our algorithm is a new notion of capped $k$-walks, which are walks of length $k$ that visit only nodes according to a fixed ordering. Our main technical contribution is an involved analysis proving several properties of such walks which may be of independent interest.Comment: To appear at STOC'1

Fast and Compact Exact Distance Oracle for Planar Graphs

For a given a graph, a distance oracle is a data structure that answers distance queries between pairs of vertices. We introduce an $O(n^{5/3})$-space distance oracle which answers exact distance queries in $O(\log n)$ time for $n$-vertex planar edge-weighted digraphs. All previous distance oracles for planar graphs with truly subquadratic space i.e., space $O(n^{2 - \epsilon})$ for some constant $\epsilon > 0$) either required query time polynomial in $n$ or could only answer approximate distance queries. Furthermore, we show how to trade-off time and space: for any $S \ge n^{3/2}$, we show how to obtain an $S$-space distance oracle that answers queries in time $O((n^{5/2}/ S^{3/2}) \log n)$. This is a polynomial improvement over the previous planar distance oracles with $o(n^{1/4})$ query time

Fast Similarity Sketching

We consider the Similarity Sketching problem: Given a universe $[u]= \{0,\ldots,u-1\}$ we want a random function $S$ mapping subsets $A\subseteq [u]$ into vectors $S(A)$ of size $t$, such that similarity is preserved. More precisely: Given sets $A,B\subseteq [u]$, define $X_i=[S(A)[i]= S(B)[i]]$ and $X=\sum_{i\in [t]}X_i$. We want to have $E[X]=t\cdot J(A,B)$, where $J(A,B)=|A\cap B|/|A\cup B|$ and furthermore to have strong concentration guarantees (i.e. Chernoff-style bounds) for $X$. This is a fundamental problem which has found numerous applications in data mining, large-scale classification, computer vision, similarity search, etc. via the classic MinHash algorithm. The vectors $S(A)$ are also called sketches. The seminal $t\times$MinHash algorithm uses $t$ random hash functions $h_1,\ldots, h_t$, and stores $\left(\min_{a\in A}h_1(A),\ldots, \min_{a\in A}h_t(A)\right)$ as the sketch of $A$. The main drawback of MinHash is, however, its $O(t\cdot |A|)$ running time, and finding a sketch with similar properties and faster running time has been the subject of several papers. Addressing this, Li et al. [NIPS'12] introduced one permutation hashing (OPH), which creates a sketch of size $t$ in $O(t + |A|)$ time, but with the drawback that possibly some of the $t$ entries are "empty" when $|A| = O(t)$. One could argue that sketching is not necessary in this case, however the desire in most applications is to have one sketching procedure that works for sets of all sizes. Therefore, filling out these empty entries is the subject of several follow-up papers initiated by Shrivastava and Li [ICML'14]. However, these "densification" schemes fail to provide good concentration bounds exactly in the case $|A| = O(t)$, where they are needed. (continued...

Practical Hash Functions for Similarity Estimation and Dimensionality Reduction

Hashing is a basic tool for dimensionality reduction employed in several aspects of machine learning. However, the perfomance analysis is often carried out under the abstract assumption that a truly random unit cost hash function is used, without concern for which concrete hash function is employed. The concrete hash function may work fine on sufficiently random input. The question is if it can be trusted in the real world when faced with more structured input. In this paper we focus on two prominent applications of hashing, namely similarity estimation with the one permutation hashing (OPH) scheme of Li et al. [NIPS'12] and feature hashing (FH) of Weinberger et al. [ICML'09], both of which have found numerous applications, i.e. in approximate near-neighbour search with LSH and large-scale classification with SVM. We consider mixed tabulation hashing of Dahlgaard et al.[FOCS'15] which was proved to perform like a truly random hash function in many applications, including OPH. Here we first show improved concentration bounds for FH with truly random hashing and then argue that mixed tabulation performs similar for sparse input. Our main contribution, however, is an experimental comparison of different hashing schemes when used inside FH, OPH, and LSH. We find that mixed tabulation hashing is almost as fast as the multiply-mod-prime scheme ax+b mod p. Mutiply-mod-prime is guaranteed to work well on sufficiently random data, but we demonstrate that in the above applications, it can lead to bias and poor concentration on both real-world and synthetic data. We also compare with the popular MurmurHash3, which has no proven guarantees. Mixed tabulation and MurmurHash3 both perform similar to truly random hashing in our experiments. However, mixed tabulation is 40% faster than MurmurHash3, and it has the proven guarantee of good performance on all possible input.Comment: Preliminary version of this paper will appear at NIPS 201

Sublinear Distance Labeling

A distance labeling scheme labels the $n$ nodes of a graph with binary strings such that, given the labels of any two nodes, one can determine the distance in the graph between the two nodes by looking only at the labels. A $D$-preserving distance labeling scheme only returns precise distances between pairs of nodes that are at distance at least $D$ from each other. In this paper we consider distance labeling schemes for the classical case of unweighted graphs with both directed and undirected edges. We present a $O(\frac{n}{D}\log^2 D)$ bit $D$-preserving distance labeling scheme, improving the previous bound by Bollob\'as et. al. [SIAM J. Discrete Math. 2005]. We also give an almost matching lower bound of $\Omega(\frac{n}{D})$. With our $D$-preserving distance labeling scheme as a building block, we additionally achieve the following results: 1. We present the first distance labeling scheme of size $o(n)$ for sparse graphs (and hence bounded degree graphs). This addresses an open problem by Gavoille et. al. [J. Algo. 2004], hereby separating the complexity from distance labeling in general graphs which require $\Omega(n)$ bits, Moon [Proc. of Glasgow Math. Association 1965]. 2. For approximate $r$-additive labeling schemes, that return distances within an additive error of $r$ we show a scheme of size $O\left ( \frac{n}{r} \cdot\frac{\operatorname{polylog} (r\log n)}{\log n} \right )$ for $r \ge 2$. This improves on the current best bound of $O\left(\frac{n}{r}\right)$ by Alstrup et. al. [SODA 2016] for sub-polynomial $r$, and is a generalization of a result by Gawrychowski et al. [arXiv preprint 2015] who showed this for $r=2$.Comment: A preliminary version of this paper appeared at ESA'1

Fast hashing with Strong Concentration Bounds

Previous work on tabulation hashing by Patrascu and Thorup from STOC'11 on simple tabulation and from SODA'13 on twisted tabulation offered Chernoff-style concentration bounds on hash based sums, e.g., the number of balls/keys hashing to a given bin, but under some quite severe restrictions on the expected values of these sums. The basic idea in tabulation hashing is to view a key as consisting of $c=O(1)$ characters, e.g., a 64-bit key as $c=8$ characters of 8-bits. The character domain $\Sigma$ should be small enough that character tables of size $|\Sigma|$ fit in fast cache. The schemes then use $O(1)$ tables of this size, so the space of tabulation hashing is $O(|\Sigma|)$. However, the concentration bounds by Patrascu and Thorup only apply if the expected sums are $\ll |\Sigma|$. To see the problem, consider the very simple case where we use tabulation hashing to throw $n$ balls into $m$ bins and want to analyse the number of balls in a given bin. With their concentration bounds, we are fine if $n=m$, for then the expected value is $1$. However, if $m=2$, as when tossing $n$ unbiased coins, the expected value $n/2$ is $\gg |\Sigma|$ for large data sets, e.g., data sets that do not fit in fast cache. To handle expectations that go beyond the limits of our small space, we need a much more advanced analysis of simple tabulation, plus a new tabulation technique that we call \emph{tabulation-permutation} hashing which is at most twice as slow as simple tabulation. No other hashing scheme of comparable speed offers similar Chernoff-style concentration bounds.Comment: 54 pages, 3 figures. An extended abstract appeared at the 52nd Annual ACM Symposium on Theory of Computing (STOC20